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By solving a set of recursion relations for the matrix elements of the ${\calU}_h(sl(2))$ generators, the finite dimensional highest weight representationsof the algebra were obtained as factor representations. Taking a nonlinearcombination of the generators of the two copies of the ${\cal U}_h(sl(2))$algebra, we obtained ${\cal U}_h(so(4))$ algebra. The latter, on contraction,yields ${\cal U}_h(e(3))$ algebra. A nonlinear map of ${\cal U}_h(e(3))$algebra on its classical analogue $e(3)$ was obtained. The inverse mapping wasfound to be singular. It signifies a physically interesting situation, where inthe momentum basis, a restricted domain of the eigenvalues of the classicaloperators is mapped on the whole real domain of the eigenvalues of the deformedoperators.Comment: 19 pages, latex-file, version to be published in Int. Jour. Mod. phy

On ${\mathcal U}_h({\rm sl}(2))$, ${\mathcal U}_h(e(3))$ and Their Representations